Geometry Chapter 6 Note-Taking Guide Name ______________________ Polygons and Parallelograms Per _____ Date ______________ 6.1 A polygon is a closed plane figure formed by three or more line segments such that: Each segment intersects exactly two other segments only at their endpoints No two segments with a common endpoint are collinear 6.1 Each segment that forms a polygon is a side of the polygon. The common endpoint of two sides is a vertex of the polygon. A segment that connects any two non-consecutive vertices is a diagonal. You can name a polygon by the number of sides: Number of sides Name of polygon 3 4 5 6 7 8 9 10 12 n 6.1 A concave polygon is a polygon is a polygon for which at least one diagonal contains points in the exterior of the polygon. A concave polygon will have at least one interior angle that measures more than 180 degrees. 6.1 A convex polygon is a polygon where no diagonal contains points in the exterior of the polygon. 6.1 A regular polygon is a convex polygon that is both equilateral and equiangular. 6.1 Determine if the following is a polygon. If it is, name the polygon by its number of sides. Determine if the polygon is concave or convex, regular or irregular. (a) (b) (c) 6.1 To find the sum of the interior angle measures of a convex polygon, draw all possible diagonals from one vertex of the polygon. This creates a set of triangles. The sum of the angles of the measures of all the triangles equals the sum of the angle measures of the polygon. 6.1 The sum of the interior angle measures of a convex polygon with n sides is SnI 2 180 6.1 If a polygon is a regular polygon, then the measure of each interior angle can be found by n 2 180 I n 6.1 (a) Find the sum of the interior angles of a (b) Find the number of sides of a polygon if the Nonagon. sum of the interior angles is 900°. 6.1 (c) Find the measure of an interior angle of a (d) Find the number of sides of a regular polygon regular hexagon. if each interior angle measures 144°. 6.1 If one exterior angle is drawn at each vertex of a polygon, the sum of the exterior angles of the convex polygon is always 360°. SE 360 6.1 If one exterior angle is drawn at each vertex of a regular polygon, the measure of each exterior angle can be calculated by 360 E n 6.1 (e) Find the sum of the exterior angles for a (f) Find the sum of the exterior angles for a triangle. dodecagon. 6.1 (g) Find the measure of an exterior angle of a (h) The measure of an exterior angle of a regular regular heptagon. polygon is 40°, find the number of sides of the polygon. 6.1 The interior and exterior angles of a polygon are Find the measure of an exterior angle of a supplementary: regular polygon if an interior angle measures EI180 60 . 6.2 A parallelogram is a quadrilateral with two pairs of parallel sides. 6.2 Properties of a parallelogram: Opposite sides are congruent Opposite angles are congruent Consecutive angles are supplementary Diagonals bisect each other. 6.2 6.2 Given: ABCD is a parallelogram Prove: AEB CED 6.3 Ways to Prove a quadrilateral is a parallelogram: 1.) Both pairs of opposite sides are parallel 2.) Both pairs of opposite sides are congruent. 3.) Both pairs of opposite angles are congruent. 4.) Diagonals bisect each other 5.) If one pair of opposite sides are BOTH congruent and parallel. 6.) One angle is supplementary to both of its consecutive angles. 6.3 6.3 Show it is a parallelogram using each of the following methods: A) slope formula: slopes are equal so opposite sides are parallel yy (slope needs to be calculated for ALL 4 sides) 21 xx21 B) midpoint formula: same midpoint means diagonals bisect each other x1 x 2 y 1 y 2 (midpoint needs to be calculated for BOTH diagonals) , 22 C) distance formula: both pairs of opposite sides are congruent 22 (distance needs to be calculated for ALL 4 sides) x2 x 1 y 2 y 1 6.3 Show that ABCD is a parallelogram by A) Opposite sides are parallel B) Diagonals bisect each other C) Opposite sides are congruent 6.4A A rectangle is a parallelogram with one right angle. 6.4A Properties of rectangles: All properties of parallelograms All angles are right angles Diagonals are congruent 6.4A 6.4A A rhombus is an equilateral parallelogram. 6.4A Properties of rhombi: All properties of parallelograms All sides are congruent Diagonals are perpendicular bisectors Diagonals bisect interior angles The four triangles formed by the diagonals are congruent. 6.4A c = ________ m MOH 74 x = _________ a = _________ OM = _______ b = __________ RM = _______ m RMO ________ HR = _______ OR = _________ 6.4A Show that A(-3,5) B(2,7) C(4,2) D(-1,0) is a rectangle. Show that ABCD is a rhombus. 6.4B A parallelogram that is both a rectangle and a rhombus is a square. It is equilateral and equiangular. 6.4B A square has all the properties of a parallelogram, a rectangle, and a rhombus. 6.4B x = ___________ y = ___________ z = ___________ a = ___________ b = ___________ 6.4B Show that A(-5,-4) B(0,2) C(6,-3) D(1,-9) is a square. 6.6 A kite is a quadrilateral with 2 pairs of disjoint (separate) consecutive sides that are congruent. The angles between the congruent sides are called vertex angles. 6.6 Properties of kites: Non-vertex angles are congruent. Diagonals are perpendicular. Diagonal connecting the non-vertex angles is bisected (divided into 2 congruent segments) Diagonal connecting the vertex angles will bisect those vertex angles. 6.6 a = _______ g = _______ b= ________ h = _______ c = _______ i = _______ d = _______ j = _______ e = _______ k = _______ f = _______ 6.6 A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called BASES. The non-parallel sides are called LEGS. BASE ANGLES are formed by the base and the legs. 6.6 An isosceles trapezoid is a quadrilateral whose legs are congruent. 6.6 Properties of an Isosceles Trapezoid: Both pairs of base angles are congruent. Diagonals are congruent. 6.6 6.6 The median of a trapezoid is segment that connects the midpoints of the legs. Also described as a MIDSEGMENT. The median of a trapezoid is parallel to the bases and measures one-half the sum of the measures of the bases. 6.6 .